This doc summarizes preliminary exploratory analysis of Fraser pink salmon spawner-recruitment relationship and associated biological benchmarks. Repository with all data and code to reproduce the analysis, and this doc, can be found here .
First, a quick look at the raw escapement and harvest time series.
Note spawning escapement has been estimated with various methods over the years which vary in their precision, and which are likely less precise than estimates of harvest. We would like to account for this time varying observation error when estimating the shape of the spawner-recruitment relationship. To do this I fit a state-space spawner recruitment model to the estimates of spawner abundance and harvest. The model is similar to the one originally described in Fleischman et al. 2013 but without age-structure given the fixed two-year pink salmon life cycle.
The model assumed a Ricker type spawner-recruitment relationship with serially correlated recruitment residuals and was parameterized with a log-normal bias correction applied directly to the likelihood for the process model. Observation error CVs on escapement that were specified for the observation model, and which are just place holders for now, were:
| Years | Method | CV |
|---|---|---|
| 1957-1985 | PSC mark-recapture (system specific) | 20% |
| 1987-1991 | DFO mark-recapture (system specific) | 20% |
| 1993-2001 | DFO mark-recapture (mainstem) | 20% |
| 2003-2007 | Test fishery | 50% |
| 2009-2011 | Mission post-season | 35% |
| 2013-2021 | Mission in-season | 35% |
And I assumed a 5% CV on harvest for all year (again as a placeholder).
I fit the spawner-recruitment model in the Stan Bayesian modelling platform, using
the rstan package.
More details on model structure, priors, diagnostics, etc. can be found
in the analysis sub-folder of the
repository.
Here are posterior distributions for a few key parameters including
magnitude of within-stock density dependence (beta), intrinsic
productivity (ln alpha), serial correlation in recruitment residuals
(phi) and magnitude of recruitment variation (sigma).
And this is what the spawner-recruitment relationship looks like.
Here are the recruitment deviations over time which suggest little
evidence for directional change in productivity, though last three brood
years have all been below average.
| median | lower.CI | upper.CI | |
|---|---|---|---|
| 80% Smsy | 4.0777 | 3.0484 | 5.931 |
| Sgen | 1.3455 | 0.6913 | 2.642 |
| Umsy | 0.5945 | 0.4679 | 0.702 |
| 25th percentile | 1.9233 | NA | NA |
| 50th percentile | 4.5600 | NA | NA |
This is what reconstructed harvest rate looks like over time relative
to \(U_{MSY}\).
And here is what reconstructed spawner abundance looks like over time
relative to upper (80% \(S_{MSY}\)) and
lower (\(S_{GEN}\)) biological
benchmarks.
We can also visualize expected yield and recruitment as a function of spawning escapement via “optimal” yield, recruitment and overfishing profiles. In panel (a) the optimal yield profile illustrates the probability that a given spawner abundance is expected to achieve 70%, 80%, or 90% of maximum sustainable yield (\(S_{MSY}\)). In panel (b) the optimal recruitment profile illustrates the probability that a given spawner abundance is expected to achieve 70%, 80%, or 90% of maximum sustainable recruitment (\(S_{MSR}\)). In panel (c) the overfishing profile illustrates the probability that sustained yield (\(SY\)) is reduced to less than a percentage (70%, 80%, or 90%) of \(MSY\) given a fixed level of escapement and is calculated as 1 – P(\(SY\) > X% of \(MSY\)) at \(S\) < \(S_{MSY}\) , and 0 at \(S\) > \(S_{MSY}\). Historic spawning escapements are shown along x-axis.
Now we’ll visualize \(U_{MSY}\) and \(S_{MSY}\) in a Kobe plot. The different quadrants represent different scenarios, where the upper left shows harvest above \(U_{MSY}\) while the stock is below \(S_{MSY}\), which is a dangerous place for a stock to be. The bottom right shows years where the stock is above \(S_{MSY}\) and harvest is below \(U_{MSY}\) which could allow the stock to be harvested more. The other quadrants represent moderately risky states for the stock to be in. The dotted vertical line shows 80% \(S_{MSY}\) (the wild salmon policy benchmark).
Initial thoughts on some next steps include: